A duality theorem related to stars

Here we present a duality theorem for dominated combs in terms of tree-decompo-sitions making the right but not the left dashed arrow in Figure 6.3.1 true.

Theorem 6.3.5. Let G be any connected graph, and let U ⊆V(G) be any vertex set. Then the following assertions are complementary:

(i) G contains a dominated comb attached to U;

(ii) G has a tree-decomposition with pairwise disjoint finite separators that dis-plays ∂ΩU.

Moreover, the tree-decomposition in (ii) can be chosen with connected separators and rooted so that it covers U cofinally.

Before we prepare the proof of our theorem, let us deduce the right dashed arrow of Figure 6.3.1 from it (also see Figure 6.3.3which shows the last two columns of Figure6.3.1 in greater detail and where Theorem6.3.5 (ii) including the theorem’s

‘moreover’ part is inserted for ‘?’): If G does not contain a star attached to U, then in particular it does not contain a dominated comb attached to U. Hence Theorem 6.3.5 yields a tree-decomposition (T,V) of Gwhich we choose so that it also satisfies the theorem’s ‘moreover’ part; in particular (T,V) is rooted so that it covers U cofinally. By assumption, the star-comb lemma yields a comb in G attached to U⁰ for every infinite subset U⁰ of U. Since (T,V) displays ∂ΩU this means that no part can meetU infinitely. And additionally employing the pairwise disjoint finite separators plus U being cofinally covered by the tree-decomposition, we deduce that no node of T can have infinite degree: Suppose for a contradiction thatt∈T is a vertex of infinite degree. For every up-neighbourt⁰ of t we choose a vertex from U that is contained in a part Vt⁰⁰ with t⁰⁰ ≥ t⁰ in T. Then applying the star-comb lemma in G to the infinitely many chosen vertices fromU yields a comb. The end of the comb’s spine must then live at t because the separators of (T,V) are all finite and pairwise disjoint. But this contradicts the fact that (T,V) displays ∂ΩU which contains the end of the comb’s spine. Finally, (T,V) inherits the properties of the ‘moreover’ part of Theorem 6.3.2 from the identical properties of Theorem 6.3.5 (ii) including that theorem’s ‘moreover’ part.

@ dominated comb

attached to U @ star attached to U

∃ tree-decomposition with (∗) that covers U cofinally

∃ locally finite tree-decomposition with all parts meetingU finitely and with (∗)

Figure 6.3.3.: The last two columns of Figure6.3.1with Theorem6.3.5(ii) including the theorem’s ‘moreover’ part inserted for ‘?’.

Condition (∗) says that the tree-decomposition displays ∂ΩU and has pairwise disjoint finite connected separators.

In order to prove Theorem6.3.5, we will employ the following result by Carmesin.

Recall that a rooted Sℵ₀-tree (T, α) has upwards disjoint separators if for every

two edges ^→e <f^→pointing away from the root r of T the separators of α(^→e) and α(f^→) are disjoint. And (T, α) is upwards connected if for every edge ^→e pointing away from the root r the induced subgraph G[B] stemming from (A, B) =α(^→e) is connected. A rooted tree-decomposition has upwards disjoint separators or is upwards connected if its correspondingSℵ0-tree is.

Theorem 6.3.6 (Carmesin 2014, Theorem 5.2.17). Every connected graph G has an upwards connected rooted tree-decomposition with upwards disjoint finite separators that displays the undominated ends of G.

Carmesin’s proof of this theorem in [19] is long and complex. However, in this chapter we need his theorem only for normally spanned graphs. This is why we will provide a substantially shorter proof for this class of graphs (cf. Theorem 6.3.10).

Furthermore, we prove that the separators of the tree-decomposition in Theo-rem 6.3.6can be chosen pairwise disjoint and connect, which makes it easier for us to apply the theorem. The latter is essentially accomplished by the following lemma:

Lemma 6.3.7. Let G be any connected graph and let Ψ be any set of ends of G.

Then the following assertions are equivalent:

(i) G has an upwards connected rooted tree-decomposition with upwards disjoint finite separators that displays Ψ;

(ii) G has a tree-decomposition with pairwise disjoint finite connected separators that displays Ψ.

Indeed, this lemma together with Theorem 6.3.6 yields the following theorem:

Theorem 6.3.8. Every connected graph G has a tree-decomposition with pairwise disjoint finite connected separators that displays the undominated ends of G.

For the proof of Lemma6.3.7we need the following lemma from the first chapter of our series:

Lemma 6.3.9 (Lemma 5.2.16). Let G be any graph. Every upwards connected rooted Sℵ₀-tree (T, α) with upwards disjoint separators displays the ends of G that correspond to the ends of T.

Proof of Lemma 6.3.7. The implication (ii)→(i) is immediate, we prove (i)→(ii).

Let (T,V) be an upwards connected rooted tree-decomposition ofGwith upwards disjoint finite separators that displays Ψ. We consider the Sℵ₀-tree (T, α) corre-sponding to (T,V). For every edgee= t1t2 ofT with t1 ≤t2 andα(t1, t2) = (A, B) we use that (T, α) is upwards connected to find a finite connected subgraph He of G[B] that contains A∩B. We define A⁰ := A∪V(He) and B⁰ :=B so that the separator A⁰∩B⁰ =V(He) is connected. Then we define α⁰(t1, t2) := (A⁰, B⁰) and α⁰(t2, t1) := (B⁰, A⁰) to obtain another map α⁰: E(T^→ )→ S^→ℵ₀. The pair (T, α⁰) does not need to be an Sℵ₀-tree, for some of its separations might cross. To fix this, we will carefully ‘thin out’ the tree and, consequently, the set of separations associated with it via α⁰. This will result in a contraction minor ˜T ofT such that ( ˜T,α˜⁰) with

˜

α⁰ :=α⁰ E( ˜T) is an Sℵ0-tree with upwards disjoint finite connected separators that still displays Ψ. Then, in order to obtain the desired tree-decomposition, we just have to contract all the edges of ˜T that are at an even distance from the root, and restrict ˜α⁰ to the smaller edge set of the resulting contraction minor of ˜T.

To begin the construction of ˜T, we partially order E(T) by letting e ≤ f whenever e precedes f on a path in T starting at the root. For every edgee of T we do the following. We writeTe for the component ofT −ethat does not contain the root. Then, we let Fe⊆E(Te) consist of the down-closure in E(Te) of those edges whose α⁰-separator (the separator of the separation that α⁰ associates with the edge) meets the α⁰-separator of e. A distance argument employing the original upwards disjointα-separators ensures thatFeinduces a rayless down-closed subtree of Te.

In order to reasonably name edges of T whose contraction leads to ˜T, we recursively construct a sequence E0, E1, . . . of pairwise disjoint subsets of E(T) such that their overall union E⁰ :=F

n∈NEn induces a partition { {e}, Fe |e∈E⁰} ofE(T). The construction goes as follows. Take E0 to be the set of minimal edges of E(T), i.e. take E0 to be the set of edges of T at the root. Then at step n >0 consider the edges of E(T) that are not contained in the down-closed edge set S{ {e}, Fe |e∈E0∪ · · · ∪En−1}, and take the minimal ones to form En.

Once we have constructedE⁰, we take ˜T to be the contraction minor of T that is obtained by contracting all the edges occurring in some Fe withe∈E⁰. Then ( ˜T,α˜⁰) has upwards disjoint finite connected separators and displays Ψ, as we verify now. Consider any distinct two edges e andf of ˜T, that is, edges e, f ∈E⁰. If the two edges are comparable withe < f, say, then theirα⁰-separators are disjoint asf is not in Fe, and so in particular their α⁰-separations are nested. Otherwise e and f are incomparable, and then their α⁰-separations are nested by the construction ofα⁰ fromα. Therefore, the separators of ( ˜T,α˜⁰) are finite, connected and pairwise disjoint. It remains to show that ( ˜T,α˜⁰) displays Ψ.

Since all Fe are rayless, we deduce that every ray of T meets E⁰ infinitely.

Consequently, the rooted rays of T correspond bijectively to the rooted rays of ˜T via the map R7→R˜ satisfying E(R)⊇E( ˜R). Now to see that ( ˜T,α˜⁰) displays Ψ, consider any end ω of G. If ω is not contained in Ψ, then ω lives at a node t∈T (with regard to (T, α)), and hence ω lives at the node ˜t∈T˜(with regard to ( ˜T,α˜⁰))

that contains t. Otherwise ω lies in Ψ. Then ω corresponds to an end of T. This end is uniquely represented by a rooted rayR of T. And then from E( ˜R)⊆E(R) it follows thatω corresponds to the end of ˜R in ˜T. So the ends in Ψ correspond to ends of ˜T while all ends in ΩrΨ live at nodes. Then by Lemma 6.3.9 this correspondence is bijective, and hence ( ˜T,α˜⁰) displays Ψ as desired.

Theorem 6.3.10. Let G be any connected graph. If T_nt ⊆ G is a normal tree such that every component ofG−T_nt has finite neighbourhood, thenGhas a rooted tree-decomposition (T,V) with the following three properties:

ˆ the separators are pairwise disjoint, finite and connected;

ˆ (T,V) displays the undominated ends in the closure of T_nt;

ˆ (T,V) covers V(T_nt) cofinally.

Proof. Given the normal tree T_nt, by Lemma 6.3.7 it suffices to find an upwards connected rooted tree-decomposition (T,V) of G that diplays the undominated ends in the closure of T_nt and that has upwards disjoint finite separators all of which meet V(T_nt).

Let us writer for the root of T_nt. Recall that every component ofG−T_nt has finite neighbourhood by assumption. Hence every end ω ∈Ωr∂ΩT_nt lives in a unique component of G−T_nt; we define the height of ω to be the height of the maximal neighbour of this component in T_nt.

Starting withT0 = randα0 =∅we recursively construct an ascending¹ sequence of Sℵ0-trees (Tn, αn) all rooted inr and satisfying the following conditions:

(i) the separators of (Tn, αn) are upwards disjoint and they are vertex sets of ascending paths inT_nt;

(ii) Tn arises fromTn−1 by adding edges to its (n−1)th level;

(iii) undominated ends in the closure ofT_nt live at nodes of thenth level of Tn

with regard to (Tn, αn);

(iv) if ω∈Ωr∂ΩT_nt has height < n, then ω lives at a node of Tn of height < n with regard to (Tn, αn).

Before pointing out the details of our construction, let us see how to complete the proof once the (Tn, αn) are defined. Consider the Sℵ0-tree (T, α) defined by letting T := S

n∈NTn and α := S

n∈Nαn, and let (T,V) be the corresponding tree-decomposition of G. By (i) we have that (T,V) is indeed a rooted tree-decomposition with upwards disjoint finite connected separators all of which meet V(T_nt). It remains to prove that (T,V) displays the undominated ends in the closure of T_nt.

By Lemma6.3.9 it suffices to show that the undominated ends in the closure of T_nt are precisely the ends of Gthat correspond to the ends of T. For the forward inclusion, consider any undominated endω in the closure ofT_nt. By (iii), it follows thatω lives at a node tn ofTn (with regard to (Tn, αn)) at level n for everyn ∈N, and these nodes form a ray R= t0t1. . . ofT. Then ω corresponds to the end of T containingR.

For an indirect proof of the backward inclusion, consider any endω of G that is either dominated or not contained in the closure of T_nt. We show that ω does not correspond to any end of T. If ω is dominated, then this follows from the fact that (T,V) has upwards disjoint finite separators. Otherwise ω is not contained in the closure of T_nt. Let n∈ N be strictly larger than the height ofω. By (iv), it follows that ω lives at a nodetω of Tn of height < nwith regard to (Tn, αn). And by (ii), the tree Tn consists precisely of the first n levels ofT. We conclude that ω lives in the part of (T,V) corresponding to tω.

Now, we turn to the construction of the (Tn, αn), also see Figure6.3.4. At step n+ 1 suppose that (Tn, αn) has already been defined and recall that the separators of (Tn, αn) are vertex sets of ascending paths in T_nt by (i). Let L be the nth level of Tn. To obtain (Tn+1, αn+1) from (Tn, αn), we will add for each `∈L new

1Here, we mean ascending in both entries with regard to inclusion, i.e., Tn ⊆ Tn+1 and αn⊆αn+1 for alln∈N.

XY Z y

Zy

Byz

y⁰

Zy⁰ =∅

Figure 6.3.4.: The construction of the (Tn, αn) in the proof of Theorem 6.3.10.

Here the vertex set Z consists of all vertices that are contained in some Zy with y ∈Y. The depicted tree isT_nt.

vertices (possibly none) to Tn that we join exactly to ` and define the image of the so emerging edges under αn+1. So fix ` ∈L. Let X be the separator of the separation corresponding to the edge between ` and its predecessor in Tn (if n= 0 put X =∅). Recall that X is the vertex set of an ascending path in T_nt by (i).

In T_nt, let Y be the set of up-neighbours of the maximal vertices in X (for n= 0 let Y := {r}). For each y ∈ Y let Zy be the set of those z ∈ bycT_nt that are minimal with the property that G contains no T_nt-path starting in dyeT_nt and ending inbzcT_nt. (Note that a normal ray of T_nt that contains y meets Zy if and only if it is not dominated by any of the vertices indyeT_nt; this fact together with (i) will guarantee (iii) for n+ 1.) Then the vertex set ofyT_ntz separates the connected sets Ayz := (V rbbzccT_nt)∪V(yT_ntz) and Byz := V(yT_ntz)∪ bbzccT_nt whenever y∈Y andz ∈Zy. Join a nodetyz to `for every pair (y, z) with y∈Y andz ∈Zy, and put αn+1(`tyz) := (Ayz, Byz). Then the Sℵ0-tree (Tn+1, αn+1) clearly satisfies (i) and (ii). That it satisfies (iii) was already argued in the construction and (iv)

follows from (i) and the definition of αn+1(`tyz).

With Theorem6.3.10 at hand, we are finally able to prove Theorem6.3.5:

Proof of Theorem 6.3.5. First, we show that (i) and (ii) cannot hold at the same time. For this, assume for a contradiction that G contains a dominated comb attached to U and has a tree-decomposition (T,V) with pairwise disjoint finite separators that displays ∂ΩU. We write ω for the end of Gcontaining the comb’s spine. Then ω lies in the closure of U, and since (T,V) displays ∂ΩU there is a unique end η of T to whichω corresponds. But as the finite separators of (T,V) are pairwise disjoint, it follows that ω is undominated inG, contradicting thatω contains the spine of a dominated comb.

Now, to show that at least one of (i) and (ii) holds, we prove ¬(i)→(ii). Using Theorem6.1we find a normal tree T_nt ⊆Gthat containsU cofinally and all whose

rays are undominated in G. Furthermore, by the ‘moreover’ part of Theorem6.1 we may assume that every component of G−T_nt has finite neighbourhood, and by Lemma 6.2.4 we have ∂ΩU = ∂ΩT_nt. Then Theorem 6.3.10 yields a rooted tree-decomposition (T⁰,V⁰) ofG as in (ii) that has connected separators and covers V(T_nt) cofinally. It remains to show that (T⁰,V⁰) can be chosen so as to cover U cofinally. For this, consider the nodes of T⁰ whose parts meetU, and let T ⊆T⁰ be induced by their down-closure in T⁰. Then let (T⁰, α⁰) be the Sℵ0-tree of G that corresponds to (T⁰,V⁰) and consider the rooted tree-decomposition (T,V) of G that corresponds to (T, α⁰ E(T^→ ) ). Now (T,V) is as in (ii) and satisfies the theorem’s ‘moreover’ part.

Im Dokument Ends and tangles, stars and combs, minors and the Farey graph (Seite 119-125)